## Nonradial solvability structure of super-diffusive nonlinear parabolic equations

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- by Panagiota Daskalopoulos and Manuel del Pino PDF
- Trans. Amer. Math. Soc.
**354**(2002), 1583-1599 Request permission

## Abstract:

We study the solvability of the Cauchy problem for the nonlinear parabolic equation \[ \frac {\partial u}{\partial t} = \mbox {div} (u^{m-1}\nabla u)\] when $m < 0$ in $\textbf {R}^2$, with $u(x,0)= f(x)$ a given nonnegative function. It is known from earlier works of the authors that the asymptotic radial growth $r^{-2/1-m}$, $r=|x|$ for the spherical averages of $f(x)$ is critical for local solvability, in particular ensuring it if $f$ is radially symmetric. We show that if the initial data $f(x)$ behaves in polar coordinates like $r^{-2/1-m} g(\theta )$, for large $r= |x|$ with $g$ nonnegative and $2\pi$-periodic, then the following holds: If $g$ vanishes on some interval of length $l^* = \frac {(m-1)\pi }{2m} >0$, then there is no local solution of the initial value problem. On the other hand, if such an interval*does not exist*, then the initial value problem is locally solvable and the time of existence can be estimated explicitly.

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## Additional Information

**Panagiota Daskalopoulos**- Affiliation: Department of Mathematics, University of California at Irvine, Irvine, California 92697
- MR Author ID: 353551
- Email: pdaskalo@math.uci.edu
**Manuel del Pino**- Affiliation: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
- MR Author ID: 56185
- Email: delpino@dim.uchile.cl
- Received by editor(s): March 25, 1998
- Published electronically: December 4, 2001
- Additional Notes: The first author was partially supported by The Sloan Foundation and by NSF/Conicyt-Chile grant INT-9802406

The second author was partially supported by grants Lineas Complementarias Fondecyt 8000010 and FONDAP - © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**354**(2002), 1583-1599 - MSC (1991): Primary 35K15, 35K55, 35K65; Secondary 35J40
- DOI: https://doi.org/10.1090/S0002-9947-01-02888-4
- MathSciNet review: 1873019